This document discusses how small catalytic P systems and purely catalytic P systems can simulate register machines to demonstrate their computational completeness. It defines these P system models and register machines. The key points are: - Small catalytic P systems use catalytic reactions and two catalysts to simulate any register machine with two decrementable registers, proving their Turing-completeness. - Purely catalytic P systems only involve catalyzed reactions and require three catalysts to simulate register machines. - The document provides examples of how ADD, SUB, and HALT instructions of a register machine can be simulated by evolution rules in catalytic P systems in different computing modes. This allows catalytic P systems to generate non-semilinear sets and demonstrate their full

1. Small(purely) Catalytic P systems simulating
register machine
Authored by
petr sosik,Miroslav langer
PRESENTED BY
MUDHIGONDA,VENKAT SAI SHARATH CHANDRA
FOR THE COURSE
CS5813 FORMAL LANGUAGES
11/22/2016

2. Table of Contents
 Introduction
 Definitions
 P systems simulation in different modes
 P systems Generating a Non-Semilineaar set
 Conclusion

3. INTRODUCTION
 P System is an abstract computing
model inspired by the behavior of
membranes in living cells, active
transport of molecules,catalyzed
reactions etc.
 Catalyst is a substance that increases
the rate of a chemical reaction
without itself undergoing any
permanent chemical change.
 Catalyst cannot change during the
reaction and they cannot travel
between regions

4. Difference between Small and purely catalytic Systems
Small Catalytic P Systems
 Small Catalytic P System consists of
difference natural behaviors including
catalytic reactions
 Two catalyst are required to prove them
as Turing –computable
Purely Catalytic P Systems
 Purely catalytic P system only involves
catalyzed reactions
 Three catalyst are required to prove them
as Turing-Computable

5. What is Register Machine?
 In mathematical logic and theoretical computer science a register machine is a
generic class of abstract machines used in a manner similar to a Turing machine.
All the models are Turing equivalent.
 In contrast to the tape and head used by a Turing machine, the model uses
multiple, uniquely addressed registers, each of which holds a single positive
integer.
 Two catalyst are enough to simulate any register machine with two decrementable
register which , in turn can compute any Turing-Computable function.

6. Basic idea
 P systems in different computing modes can only compute semilinear sets
 Non-Semilinear set is set of integers or tuple of integers constructed in such a way
that they do not follow any sequence(like arthmetic progression)
 The P system with catalyst proved to have computational completeness but there
was no example of cataysts generating a non- Semilinear set

7. Definitions
 By N we denote the set of all nonnegative integers, NRE denotes the class of all
recursively enumerable sets of non negative integers
 PsRE the class of all recursively enumerable vectors of nonnegative integers. For
a finite set of symbolsV, by V∗
 we denote the set of all strings consisting of letters from V, including the empty
string.
 Following the notation , for integers m ≥2and k, lsuch that 1 ≤k, l ≤m, we define

8. Definitions
 A register machine is a tuple M=(m,H,lo,lh,I), where:
• m is the number of registers;
• H is the set of labels of instructions;
• lo is the label of the initial instruction;
• lh is the label of the final (halting) instruction;
• I is the program, i.e., a set of instructions labeled in a one-to-one manner by the
elements of H.

9. Register Instructions
 The instruction types are:
• (ADD(r),i,j)– add 1 to the contents of register r and choose non-deterministically
to continue with the instruction labeled i or j ; if the machine is deterministic, then
the instruction adopts the form (ADD(r),i);
• (SUB(r),i,j)– if the contents of the register r is nonzero, then subtract 1 from it and
continue with instruction i, else continue with instruction j;
• HALT– halt the machine.

10. Catalytic P system
 Acatalytic P systemof degree m ≥1is a construct
• 𝝅 =(O,C,μ,w1,...,wm,R1,...,Rm,i0)
• where:
• O is the alphabet of objects;
• C ⊆ Ois the alphabet of catalysts;
• μ is a membrane structure of degree m with membranes labeled in a one-to-one manner
with the natural numbers 1, 2, ..., m;
• w1, ..., wm ∈ O∗ are strings representing multisets of objects initially present in the m
regions of μ;
• Ri, 1 ≤i ≤m, are finite sets of evolution rules over O associated with the regions 1, 2, ...,
mof μ; these evolution rules are of the forms ca →cv or a →v, where c is a catalyst, a is an
object from O −C, and vis a string from ((O −C) ×{here, out, in})∗;
• I0 ∈ {0, 1, ..., m} indicates the output region of 𝝅.

11. Catalytic P Systems in Generating mode
 Here # is called trap symbol
 # is used to when two objects are in
same state and need a catalyst to
complete the reaction
 # is also used when an wrong step is
chosen non deterministically such that
the process never halts
Rules for trap
symbol
Rules for add
instruction
Rules for sub
instruction

12. Simulation of ADD instruction
 An ADD-instruction j :(ADD(r), k, l) ∈ I is simulated by the no n-deterministic
choice of one of the rules
 c1p̃j→c1pkd2or
 c1p̃j→c1pld2or
 Simultaneously, the rule c2d→c2is executed in order to keep the catalyst c2busy.
 If this catalyst instead acts in the rule c2o2→c2d2 then in the next step there are
two objects d2and at least one is subject to the rule d2→#, introducing the trap
symbol #.

13. simulation of SUB instructions

14. Simulation of HALT instruction
 The instruction lh:(HALT)is simulated without using any special objects or rules
of the P system. We simply modify each previously described rule producing the
symbol plhby replacing this object at the right-hand side of the rule by d1. When
such a rule is executed, the P systems finally halts with both simulated registers
r1and r2being empty

15. Result
 Cataltyic P systems halt these instructions after following rules
 |R| ≤ 3nA + 6nS + 11 for catalytic Psystems,
 |R| ≤ 3nA + 6nS + 13 for purely catalyticP systems
• where nA and nS is the number of instructions ADD and SUB, respectively

16. Catalytic P Systems in Computing mode
 In this mode there are ‘r’ number of catalysts
which can decrement with associated r
registers
 Here ADD instructions are simulated
deterministically
 j : (ADD(r),k) ∈ I
 SUB instructions are guessed non-
determinstically

17. Simulation of
SUB instructions
 When the wrong step is chosen
non-deterministically # is
introduced
 The correct steps of simulation
results in halting of instructions else
the computation cannot produce
any result
 If there are any undesired rules
catalysts are kept busy in other
auxiliary rules

18. Result
 Catalytic P system can achieve its result in following rules
 |R| ≤ 2nA+6nS+5m+1.
 By using generalized register machine we can reduce the result to
 |R| ≤ 6nS + 5m +1 for catalytic P systems,
 |R| ≤ 6nS + 6m + 1 forpurely catalytic P systems.
 In similar manner Catalytic P systems in accepting mode can be achieved in
following rules
 |R| ≤ 2nA+6nS+5m+1.

19. Catalytic P systems
generating Semilinear
sets
 Consider a non-deterministic register
machine with three registers generating the
set {2n −2n | n ≥2}. The machine starts
with all registers empty and it runs the
following program which stores the result
in register 3

20. Catalytic P systems generating
Semilinear sets
 The register machine program described works as
follows:
• The contents of register 1 is emptied and duplicated
to register 2
• then the contents of register 2 is copied to register 1
and added to register 3 .
• Finally, the only non-deterministic instruction ADD
labeled 7 increments register 1 and it decides non-
deterministically whether the computation continues
or whether it halts.
• The whole cycle is repeated until a jump to the
instruction HALT is chosen non-deterministically.

21. GENERALISED REGISTER MACHINE INSTRUCTIONS
 we can express the program given above with generalized SUB-instructions
1: (SUB(1),ADD(2)ADD(2) 1,2)
2: (SUB(2),ADD(1)ADD(3)2,ADD(1)3)
3: HALT

22. CONCLUSION
 The paper was hard to understand in the beginning. So, I have started reading the
relevant papers to get idea of P systems
 This paper proves Universal Completeness of P systems by computing non-
semilinear sets
 The author utilizes catalyst which are equal to no off register machines in
computing mode but the paper states that they use only two catalysts